Myerson and Satterthwaite [1]} consider distributions \({\mathcal {F}^H_s},{\mathcal {F}^H_b}\)  that are supported on \([\underline{s},\overline{s}]\)  and \([\underline{b},\overline{b}]\) , respectively, and have continuous and positive density everywhere in their domains.
They prove that if \(\underline{b}<\overline{s}\)  then any Bayesian incentive compatible and interim individually rational mechanism that is also ex-ante budget balanced (i.e., the expected payment that the buyer pays is at least the expected amount that the seller receives) never guarantees full efficiency (optimal GFT). They provide a formula for computing the mechanism that maximizes the GFT among all incentive compatible, individually rational and budget-balanced mechanisms (the second-best mechanism).
